Topic 3 Sets, logic and probability - kennso - home · PDF file · 2011-04-19Topic 3 Sets, logic and probability Syllabus content ... 2 A is the set of whole numbers between 50 and

3 Sets, logic and probabilityTop

ic

Syllabus content 3.1 Basic concepts of set theory: subsets; intersection; union; complement.

3.2 Venn diagrams and simple applications.

3.3 Sample space: event, A; complementary event, A′.

3.4 Basic concepts of symbolic logic: definition of a proposition; symbolicnotation of propositions.

3.5 Compound statements: implication, ⇒; equivalence, ⇔; negation, ¬ ;conjunction, ∧; disjunction, ∨; exclusive disjunction, ∨–.

Translation between verbal statements, symbolic form and Venndiagrams.

Knowledge and use of the ‘exclusive disjunction’ and the distinctionbetween it and ‘disjunction’.

3.6 Truth tables: the use of truth tables to provide proofs for the properties of connectives; concepts of logical contradiction and tautology.

3.7 Definition of implication: converse; inverse; contrapositive.Logical equivalence.

3.8 Equally likely events.

Probability of an event A given by P(A) = .

Probability of a complementary event, P(A′) = 1 − P(A).

3.9 Venn diagrams; tree diagrams; tables of outcomes. Solutions of problems using ‘with replacement’ and ‘without replacement’.

3.10 Laws of probability.

Combined events:

P(A � B) = P(A) + P(B) − P(A � B).

Mutually exclusive events:

P(A � B) = P(A) + P(B).

Independent events: P(A � B) = P(A)P(B).

Conditional probability: P(A⎥ B) = .P(A � B)P(B)

n(A)n(U)

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IntroductionIf the three areas of sets, logic and probability are looked at from a historicalperspective, then logic came first. The study of logic developed in China, Indiaand Greece, each independently of the other two, in the fourth century BC.

In the seventeenth century Pascal and others began to study probability. Thestudy of sets did not truly begin until around 1900 when Georg Cantor and RichardDedekind began work on the theory of sets.

3.1 Set theoryThe modern study of set theory began with Georg Cantor and Richard Dedekind in an1874 paper titled ‘On a characteristic property of all real algebraic numbers’. It is mostunusual to be able to put an exact date to the beginning of an area of mathematics.

The language of set theory is the most common foundation to all mathematicsand is used in the definitions of nearly all mathematical objects.

A set is a well-defined group of objects or symbols. The objects or symbols arecalled the elements of the set. If an element e belongs to a set S, this is representedas e � S. If e does not belong to set S this is represented as e � S.

1 A particular set consists of the following elements:{South Africa, Namibia, Egypt, Angola, …}.

a Describe the set.b Add another two elements to the set.c Is the set finite or infinite?

a The elements of the set are countries of Africa.b e.g. Zimbabwe, Ghanac Finite. There is a finite number of countries in Africa.

2 Consider the set{1, 4, 9, 16, 25, …}.a Describe the set.b Write another two elements of the set.c Is the set finite or infinite?

a The elements of the set are square numbers.b e.g. 36, 49c Infinite. There is an infinite number of square numbers.

Set theory 63

Georg Cantor

Worked examples


64 SETS, LOGIC AND PROBABILITY

� Exercise 3.1.11 For each of the following sets:

i) describe the set in wordsii) write down another two elements of the set.

a {Asia, Africa, Europe, …}b {2, 4, 6, 8, …}c {Sunday, Monday, Tuesday, …}d {January, March, July, …}e {1, 3, 6, 10, …}f {Mehmet, Michael, Mustapha, Matthew, …}g {11, 13, 17, 19, …}h {a, e, i, …}i {Earth, Mars, Venus, …}j A = {x|3 ≤ x ≤ 12}k S = {y| −5 ≤ y ≤ 5}

2 The number of elements in a set A is written as n(A).Give the value of n(A) for the finite sets in question 1 above.

SubsetsIf all the elements of one set X are also elements of another set Y, then X is said tobe a subset of Y.

This is written as X � Y.

If a set A is empty (i.e. it has no elements in it), then this is called the empty setand it is represented by the symbol ∅. Therefore A = ∅.

The empty set is a subset of all sets. For example, three girls, Winnie, Natalieand Emma, form a set A.

A = {Winnie, Natalie, Emma}All the possible subsets of A are given below:B = {Winnie, Natalie, Emma}C = {Winnie, Natalie}D = {Winnie, Emma}E = {Natalie, Emma}F = {Winnie}G = {Natalie}H = {Emma}I = ∅

Note that the sets B and I above are considered as subsets of A,

i.e. A � A and ∅ � A.

However, sets C, D, E, F, G and H are considered proper subsets of A. Thisdistinction in the type of subset is shown in the notation below. For proper subsets, we write:

C � A and D � A etc. instead of C � A and D � A.

Similarly G � H implies that G is not a subset of HG H implies that G is not a proper subset of H.


A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}a List the subset B of even numbers.b List the subset C of prime numbers.

a B = {2, 4, 6, 8, 10}b C = {2, 3, 5, 7}

� Exercise 3.1.21 P is the set of whole numbers less than 30.

a List the subset Q of even numbers.b List the subset R of odd numbers.c List the subset S of prime numbers.d List the subset T of square numbers.e List the subset U of triangular numbers.

2 A is the set of whole numbers between 50 and 70.a List the subset B of multiples of 5.b List the subset C of multiples of 3.c List the subset D of square numbers.

3 J = {p, q, r}a List all the subsets of J.b List all the proper subsets of J.

4 State whether each of the following statements is true or false.a {Algeria, Mozambique} � {countries in Africa}b {mango, banana} � {fruit}c {1, 2, 3, 4} � {1, 2, 3, 4}d {1, 2, 3, 4} � {1, 2, 3, 4}e {volleyball, basketball} � {team sport}f {4, 6, 8, 10} {4, 6, 8, 10}g {potatoes, carrots} � {vegetables}h {12, 13, 14, 15} {whole numbers}

The universal setThe universal set (U) for any particular problem is the set which contains all thepossible elements for that problem.

The complement of a set A is the set of elements which are in U but not in A.The set is identified as A′. Notice that U′ = ∅ and ∅′ = U.

1 If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 2, 3, 4, 5}, what set is representedby A′?

A′ consists of those elements in U which are not in A.Therefore A′ = {6, 7, 8, 9, 10}.

2 If U is the set of all three-dimensional shapes and P is the set of prisms, what setis represented by P′?

P′ is the set of all three-dimensional shapes except prisms.

Set theory 65

Worked example

Worked examples



Intersections and unionsThe intersection of two sets is the set of all the elements that belong to both sets.The symbol � is used to represent the intersection of two sets.

If P = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and Q = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} then P � Q = {2, 4, 6, 8, 10} as these are the numbers that belong to both sets.

The union of two sets is the set of all elements that belong to either or both setsand is represented by the symbol �.

Therefore in the example above, P � Q = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20}.

Unions and intersections of sets can be shown diagrammatically using Venn diagrams.

3.2 Venn diagramsVenn diagrams are the principal way of showing sets diagrammatically. They arenamed after the mathematician John Venn (1834–1923). The method consistsprimarily of entering the elements of a set into a circle or circles.

Some examples of the uses of Venn diagrams are shown below.

A = {2, 4, 6, 8, 10} can be represented as:

Elements which are in more than one set can also be represented using a Venndiagram.

P = {3, 6, 9, 12, 15, 18} and Q = {2, 4, 6, 8, 10, 12} can be represented as:

The elements which belong to both sets are placed in the region of overlap of thetwo circles.

A

2

10

8

6

4

P

6

Q

12

1018

15 8

4

29

3


Venn diagrams 67

As mentioned in the previous section, when two sets P and Q overlap as they doabove, the notation P � Q is used to denote the set of elements in the intersection,i.e. P � Q = {6, 12}. Note that 6 � P � Q; 8 � P � Q.

J = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100} and K = {60, 70, 80} can be representedas shown below; this is shown in symbols as K � J.

X = {1, 3, 6, 7, 14} and Y = {3, 9, 13, 14, 18} are represented as:

The union of two sets is everything which belongs to either or both sets and is represented by the symbol �. Therefore, in the example above, X � Y = {1, 3, 6, 7, 9, 13, 14, 18}.

� Exercise 3.2.11 Using the Venn diagram, indicate whether the following statements are true or

false. � means ‘is an element of’ and � means ‘is not an element of’.

a 5 � A b 20 � Bc 20 � A d 50 � Ae 50 � B f A � B = {10, 20}

J

K

60

70

80

1020

30

40

50

90 100

X

3

Y

14

9

7

1

13

18

6

A

10

B

20

30

15

5

40

50


2 Copy and complete the statement A � B = {...} for each of the Venn diagramsbelow.

3 Copy and complete the statement A � B = {...} for each of the Venn diagramsin question 2 above.

4 Using the Venn diagram, copy and complete these statements.

a U = {...}b A′ = {...}

5 Using the Venn diagram, copy and complete the following statements.

a U = {...}b A′ = {...}c A � B = {...}d A � B = {...}e (A � B)′ = {...}f A � B′ = {...}

6

a Using the Venn diagram, describe in words the elements of:i) set A ii) set B iii) set C.

b Copy and complete the following statements.i) A � B = {...} ii) A � C = {...} iii) B � C = {...}iv) A � B � C = {...} v) A � B = {...} vi) C � B = {...}


A

4

B

6

9

8

2

13

18

10

3

a A

4

B

9

51

7

8

166

b A BRed

Orange

Violet

Indigo

Blue

Pink

Purple

Green

Yellow

c

Ap

ts

r

qa

U

b

A

U

B

6

7

5

3

2 8

4

1

A B

C

2

15

9

3

2016

84

12

610

14


Venn diagrams 69

7

a Using the Venn diagram, copy and complete the following statements.i) A = {...} ii) B = {...} iii) C′ = {...}iv) A � B = {...} v) A � B = {...} vi) (A � B)′ = {...}

b State, using set notation, the relationship between C and A.

8

a Copy and complete the following statements.i) W = {...} ii) X = {...} iii) Z′ = {...}iv) W � Z = {...} v) W � X = {...} vi) Y � Z = {...}

b Which of the named sets is a subset of X?

9 A = {Egypt, Libya, Morocco, Chad}B = {Iran, Iraq, Turkey, Egypt}

a Draw a Venn diagram to illustrate the above information.b Copy and complete the following statements.

i) A � B = {...} ii) A � B = {...}

10 P = {2, 3, 5, 7, 11, 13, 17}Q = {11, 13, 15, 17, 19}

a Draw a Venn diagram to illustrate the above information.b Copy and complete the following statements.

i) P � Q = {...} ii) P � Q = {...}

11 B = {2, 4, 6, 8, 10}A � B = {1, 2, 3, 4, 6, 8, 10}A � B = {2, 4}Represent the above information on a Venn diagram.

12 X = {a, c, d, e, f, g, l}Y = {b, c, d, e, h, i, k, l, m}Z = {c, f, i, j, m}Represent the above information on a Venn diagram.

A B

C

2 9

87

6

5

43

1

W X

Y

210

9

8

7

6

5

4 3

1

Z



13 P = {1, 4, 7, 9, 11, 15}Q = {5, 10, 15}R = {1, 4, 9}Represent the above information on a Venn diagram.

Commutative, associative and distributive properties of setsSet A = {2, 3, 4}, B = {1, 3, 5, 7} and C = {3, 4, 5}.

A � B = B � A = {1, 2, 3, 4, 5, 7}A � B = B � A = {3}

Therefore the union and intersection of sets are commutative (the same whicheverway round the sets are ordered).

If C is added to the Venn diagram, we get:

(A � B) � C = A � (B � C) = {3}(A � B) � C = A � (B � C) = {1, 2, 3, 4, 5, 7}

therefore the union and intersection of sets are associative (the order of theoperations does not matter).

From the Venn diagram above it can also be seen that:

A � (B � C) = (A � B) � (A � C) = {2, 3, 4, 5}A � (B � C) = (A � B) � (A � C) = {3, 4}.

Therefore the union is distributive over the intersection of sets.

A

3

B

1

4

2

5

7

A B

C

2

7

1

4 5

3


Problems involving sets

1 In a class of 31 students, some study physics and some study chemistry. If 22study physics, 20 study chemistry and 5 study neither, calculate the number ofstudents who take both subjects.

The information given above can be entered in a Venn diagram in stages.The students taking neither physics nor chemistry can be put in first (as shown).

This leaves 26 students to be entered into the set circles.

If x students take both subjects then:

n(P) = 22 − x + xn(C) = 20 − x + xP � C = 31 − 5 = 26

Therefore 22 − x + x + 20 − x = 2642 − x = 26

x = 16Substituting the value of x into the Venn diagram gives:

Therefore the number of students taking both physics and chemistry is 16.

2 In a region of mixed farming, farms keep goats, cattle or sheep. There are 77farms altogether. 19 farms keep only goats, 8 keep only cattle and 13 keep onlysheep. 13 keep both goats and cattle, 28 keep both cattle and sheep and 8 keepboth goats and sheep.

a Draw a Venn diagram to show the above information.b Calculate n(G � C � S).

Venn diagrams 71

P C

5

U

P C

5

U

20 �xx22 �x

P C

5

6 16 4

U

Worked examples


First of all draw a partlycomplete Venn diagram,filling in some of theinformation above.

We know that:.

n(Only G) + n(Only C) + n(Only S) = 19 + 8 + 13 = 40

So the number of farms that keep two or more types of animal is 77 − 40 = 37.So, if n(G � C � S) = x (i.e. x is the number of farms keeping cattle, sheepand goats), then

13 − x + 8 − x + 28 − x + x = 3749 − 2x = 3749 − 37 = 2x

6 = x

It is then easy to complete the Venn diagram as shown:

b As worked out in part a and shown in the diagram, n(G � C � S) = 6.

� Exercise 3.2.21 In a class of 35 students, 19 take Spanish, 18 take French and 3 take neither.

Calculate how many take:a both French and Spanishb just Spanishc just French.

2 In a year group of 108 students, 60 liked football, 53 liked tennis and 10 likedneither. Calculate the number of students who liked football but not tennis.


G C

S

19 8

13

G C

S

19 7

62 22

8

13


3 In a year group of 113 students, 60 liked hockey, 45 liked rugby and 18 likedneither. Calculate the number of students who:

a liked both hockey and rugbyb liked only hockey.

4 One year, 37 students sat an examination in physics, 48 sat an examination inchemistry and 45 sat an examination in biology. 15 students sat examinations inphysics and chemistry, 13 sat examinations in chemistry and biology, 7 satexaminations in physics and biology and 5 students sat examinations in all three.

a Draw a Venn diagram to represent this information.b Calculate n(P � C � B).c Calculate n(P � C).d Calculate n(B � C).e How many students took an examination in only one subject?

5 On a cruise around the coast of Turkey, there are 100 passengers and crew. Theyspeak Turkish, French and English.

Out of the total of 100, 14 speak all three languages, 18 speak French andTurkish only, 16 speak English and French only, and 10 speak English andTurkish only.

Of those speaking only one language, the number speaking only French or onlyEnglish is the same and 6 more than the number that speak only Turkish.

a How many speak only French?b How many speak only Turkish ?c In total, how many speak English?

6 In a group of 125 students who play tennis, volleyball or football, 10 play allthree. Twice as many play tennis and football only. Three times as many playvolleyball and football only, and 5 play tennis and volleyball only.

If x play tennis only, 2x play volleyball only and 3x play football only, work out:

a how many play tennisb how many play volleyballc how many play football.

3.3 Sample spaceSet theory can be used to study probability.

A sample space is the set of all possible results of a trial or experiment. Eachresult or outcome is sometimes called an event.

Complementary eventsA dropped drawing pin can land either pin up, U, or pin down, D. These are theonly two possible outcomes and cannot both occur at the same time. The twoevents are therefore mutually exclusive (cannot happen at the same time) andcomplementary (the sum of their probabilities equal 1). The complement of anevent A is written A′.

Therefore P(A) + P(A′) = 1. In words this is read as ‘the probability of event Ahappening added to the probability of event A not happening equals 1’.

Sample space 73


1 A fair dice is rolled once. What is its sample space?The sample space S is the set of possible outcomes or events. ThereforeS = {1, 2, 3, 4, 5, 6} and the number of outcomes or events is 6.

2 a What is the sample space, S, for two drawing pins dropped together.b How many possible outcomes are there?

a S = {UU, UD, DU, DD}b There are four possible outcomes.

3 The probability of an event B happening is P(B) = . Calculate P(B′).

P(B) and P(B′) are complementary events, so P(B) + P(B′) = 1.

P(B′) = 1 − = .

� Exercise 3.3.11 What is the sample space and the number of events when three coins are tossed?

2 What is the sample space and number of events when a blue dice and a red diceare rolled? (Note: (1, 2) and (2, 1) are different events.)

3 What is the sample space and the number of events when an ordinary dice isrolled and a coin is tossed?

4 A mother gives birth to twins. What is the sample space and number of events fortheir sex?

5 What is the sample space if the twins in question 4 are identical?

6 Two women take a driving test.a What are the possible outcomes?b What is the sample space?

7 A tennis match is played as ‘best of three sets’.a What are the possible outcomes?b What is the sample space?

8 If the tennis match in question 7 is played as ‘best of five sets’,a what are the possible outcomes?b what is the sample space?

3.4 LogicIn philosophy, traditional logic began with the Greek philosopher Aristotle. His sixtexts are collectively known as The Organon. Two of them, Prior Analytics and DeInterpretatione, are the most important for the study of logic.

The fundamental assumption is that reasoning (logic) is built from propositions.A proposition is a statement that can be true or false. It consists of two terms: oneterm (the predicate) is affirmed (true) or denied (false) by the other term (thesubject): for example

‘All men [subject] are mortal [predicate].’

25

35

35


Worked examples

Aristotle 384–322 BCE


Logic 75

There are just four kinds of proposition in Aristotle’s theory of logic.

A type: Universal and affirmative − ‘All men are mortal.’I type: Particular and affirmative − ‘Some men are philosophers.’E type: Universal and negative − ‘No men are immortal.’O type: Particular and negative − ‘Some men are not philosophers.’

This is the fourfold scheme of propositions. The theory is a formal theoryexplaining which combinations of true premises give true conclusions.

A century later, in China, a contemporary of Confucius, Mozi ‘Father Mo’ (430 BC), is credited with founding the Mohist school of philosophy, which studied ideas of valid inference and correct conclusions.

What is logic? Logic is a way to describe situations or knowledge that enables us to reason fromexisting knowledge to new conclusions. It is useful in computers and artificialintelligence where we need to represent the problems we wish to solve using asymbolic language.

Logic, unlike natural language, is precise and exact. (It is not always easy tounderstand logic, but it is necessary in a computer program.) An example of alogical argument is:

All students are poor.I am a student.By using logic, it follows that I am poor.

Note: if the original statement is false, the conclusion is still logical, even though it isfalse, e.g.

All students are rich. (is not true)I am a student.By using logic it follows that I am rich!

It is not the case that all students are rich but, if it were, I would be rich because Iam a student. This is why computer programmers talk of ‘Garbage in, garbage out’.

Logic systems are already in use for such things as the wiring systems of aircraft.The Japanese are using logic experiments with robots.

� Exercise 3.4.11 You have four letters. A letter can be sent sealed or open. Stamps are either 10

cents or 15 cents.

Which envelope must be turned over to test the rule ‘If a letter is sealed, it musthave a 15 cent stamp’?

Unsealed

A

Sealed

B C

15c 10c

D


2 You have four cards.A letter A−Z is on one side.A number 0−9 is on the other side.You have these cards:

Which card do you turn over to test the rule ‘If a card has a vowel on one side, itmust have an even number on the other side’?

3.5 Sets and logical reasoningProposition: A proposition is a stated fact. It may also be called a statement. It can be true or

false. For example,

‘Nigeria is in Africa’ is a true proposition.‘Japan is in Europe’ is a false proposition. These are examples of simple propositions.

Compound statement: Two or more simple propositions can be combined to form a compoundproposition or compound statement.

Conjunction: Two simple propositions are combined with the word and, e.g.

p: Japan is in Asia. q: The capital of Japan is Tokyo.

These can be combined to form: Japan is in Asia and the capital of Japan isTokyo.

This is written p ∧ q, where ∧ represents the word and.

Negation: The negation of any simple proposition can be formed by putting ‘not’ into thestatement, e.g.

p: Ghana is in Africa. q: Ghana is not in Africa.

Therefore q = ¬ p (i.e. p is the negation of q).

If p is true then q cannot also be true.

Implication: For two simple propositions p and q, p ⇒ q means if p is true then q is also true,e.g.

p: It is raining. q: I am carrying an umbrella.

Then p ⇒ q states: If it is raining then I am carrying an umbrella.

Converse: This is the reverse of a proposition. In the example above the converse of p ⇒ qis q ⇒ p. Note, however, although p ⇒ q is true, i.e. If it is raining then I mustbe carrying an umbrella, its converse q ⇒ p is not necessarily true, i.e. it is notnecessarily the case that: If I am carrying an umbrella then it is raining.


E K 4 7


Sets and logical reasoning 77

Equivalent If two propositions are true and converse, then they are said to be equivalent. propositions: For, example if we have two propositions

p: Pedro lives in Madrid.q: Pedro lives in the capital city of Spain.

these propositions can be combined as a compound statement:

If Pedro lives in Madrid, then Pedro lives in the capital city of Spain.

i.e. p implies q (p ⇒ q)

This statement can be manipulated to form its converse:

If Pedro lives in the capital of Spain, then Pedro lives in Madrid.

i.e. q implies p (q ⇒ p)

The two combined statements are both true and converse so they are said to belogically equivalent (q ⇔ p). Logical equivalence will be discussed further, laterin this section.

Disjunction: For two propositions, p and q, p ∨ q means either p or q is true or both are true, e.g.

p: It is sunny. q: I am wearing flip-flops.

Then p ∨ q states either it is sunny or I am wearing flip-flops or it is both sunnyand I am wearing flip-flops.

Exclusive disjunction: For two propositions, p and q, p ∨– q means either p or q is true but not both aretrue, e.g.

p: It is sunny. q: I am wearing flip-flops.

Then p ∨– q states either it is sunny or I am wearing flip-flops only.

Valid arguments: An argument is valid if the conclusion follows from the premises (the statements).A premise is always assumed to be true, even though it might not be, e.g.

London is in France. the first premiseFrance is in Africa. the second premiseTherefore London is in Africa. the conclusion

In this case, although both premises and the conclusion are false, the argument islogically valid.

The validity of an argument can be tested usingVenn diagrams.

If p, q and r are three statements and if p ⇒ q and q ⇒ r, then it follows that p ⇒ r.In terms of sets, if A, B and C are all propersubsets (�) of the universal set U and if A � Band B � C then A � C.

Diagrammatically this can be represented as shown in the Venn diagram opposite:

A

BC U


� Exercise 3.5.11 Which of the following are propositions?

a Are you from Portugal? b Capetown is in South Africa.c Catalan is a Spanish language. d Be careful with that.e x = 3 f x ≠ 3g I play football. h Go outside and play.i Apples are good to eat. j J is a letter of the alphabet.

2 Form compound statements using the word ‘and’ from the two propositions givenand say whether the compound statement is true or false.

a t: Teresa is a girl. a: Abena is a girl.b p: x < 8 q: x > −1c a: A pentagon has 5 sides. b: A triangle has 4 sides.d l: London is in England. e: England is in Europe.e k: x < y l: y < zf m: 5 � {prime number} n: 4 � {even numbers} g s: A square is a rectangle. t: A triangle is a rectangle.h p: Paris is the capital of France. g: Ghana is in Asia.i a: 37 � {prime number} b: 51 � {prime numbers} j p: parallelograms � {rectangles} t: trapeziums � {rectangles}

The analogy of logic and set theoryThe use of No or Never or All … do not in statements (e.g. No French people areBritish people) means the sets are disjoint, i.e. they do not overlap.

The use of All or If … then or No … not in statements (e.g. There is no nursewho does not wear a uniform) means that one set is a subset of another.

The use of Some or Most or Not all in statements (e.g. Some televisions arevery expensive) means that the sets intersect.

1 P is the set of French people and Q is the set of British people. Draw a Venn diagram to represent the sets.

The Venn diagram is as shown, i.e. P � Q = ∅In logic this can be written p ∨– q, i.e. p or q but not both.

2 P is the set of nurses and Q is the set of people who wear uniform. Draw a Venn diagram to represent the sets.

P is a subset of Q as there are other people who wear uniforms apart from nurses, i.e. P � QIn logic this can be written p ⇒ q.


P Q

Worked examples

PQ


3 P is the set of televisions and Q is the set ofexpensive electrical goods. Draw a Venn diagramto represent the sets.

P intersects Q as there are expensive electricalgoods that are not televisions and there aretelevisions that are not expensive.In logic the intersection can be written p ∧ q.

� Exercise 3.5.21 Illustrate the following sets using a Venn diagram.

Q: students wearing football shirtsP: professional footballers wearing football shirts

Shade the region that represents the statement ‘Kofi is a professional footballerand a student’. How would you write this using logic symbols?

2 Illustrate the following sets using a Venn diagram.Q: students wearing football shirtsP: professional footballers wearing shirts

Shade the region that satisfies the statement ‘Maanu is either a student or aprofessional footballer but he is not both’. How would you write this using logicsymbols?

3 Illustrate the following sets using a Venn diagram.P: maths students U: all students.

Shade the region that satisfies the statement ‘Boamah is not a maths student’.How would you write this using logic symbols?

4 Illustrate the following sets using a Venn diagram.P: five-sided shapes U: all shapes

Shade the region that satisfies the statement ‘A regular pentagon is a five-sidedshape’. How would you write this using logic symbols?

5 Illustrate the following sets using a Venn diagram. Q: multiples of 5 U: integers

Shade the region where you would place 17. How would you write this using logicsymbols?

6 Illustrate the following sets using a Venn diagram. P: people who have studied medicine Q: people who are doctors

Shade the region that satisfies the statement ‘All doctors have studied medicine’.How would you write this using logic symbols?

7 Illustrate the statement people with too much money are never happy using aVenn diagram with these sets.

P: people who have too much moneyQ: people who are happy

Shade the region that satisfies the statement ‘People with too much money arenever happy’. How would you write this using logic symbols?

Sets and logical reasoning 79

P Q


8 Illustrate the following sets using a Venn diagram. P: music lessons Q: lessons that are expensive

Shade the region that satisfies the statement ‘Some music lessons are expensive’.How would you write this using logic symbols?

3.6 Truth tablesIn probability experiments, a coin when tossed can land on heads or tails. These arecomplementary events, i.e. P(H) + P(T) = 1.

In logic, if a statement is not uncertain, then it is either true (T) or false (F). Ifthere are two statements, then either both are true, both are false or one is true andone is false.

A truth table is a clear way of showing the possibilities of statements. Let proposition p be ‘Coin A lands heads’ and proposition q be ‘Coin B lands

heads’. The truth table below shows the different possibilities when the two coinsare tossed. Alongside is a two-way table also showing the different outcomes. Notethe similarity between the two tables.

Truth table Two-way table

Symbols used in logicThere are some symbols that you will need to become familiar with when we studylogic in more detail.

The following symbols refer to the relationship between two propositions pand q.

Symbol Meaning

∧ p and q (conjunction)

∨ p or q or both (inclusive disjunction)

∨− p or q but not both (exclusive disjunction)

⇒ If p then q (implication)

⇔ If p ⇒ q and q ⇒ p the statements are equivalent, i.e. p ⇔ q (equivalence)

¬ If p is true, q cannot be true. p ¬ q (negation)


p q

T T

T F

F T

F F

Coin A Coin B

H H

H T

T H

T T


Conjunction, disjunction and negationExtra columns can be added to a truth table.

p ∧ q (conjunction) means that both p and q must be true for the statement to be true.

p ∨ q (inclusive disjunction) means that either p or q, or both, must be true for the statement to be true.

p ∨− q (exclusive disjunction) means that either p or q, but not both, must be true for the statement to be true.

¬ p represents a negation, i.e. p must not be true for the statement to be true.

� Exercise 3.6.11 Copy and complete the truth table for three propositions

p, q and r. It may help to think of spinning three coins and drawing a table of possible outcomes.

Truth tables 81

p q p ∧ q

T T T

T F F

F T F

F F F

p q p ∨ q

T T T

T F T

F T T

F F F

p q p ∨_ q

T T F

T F T

F T T

F F F

p q ¬ p

T T F

T F F

F T T

F F T

p q r

T T T

F F F


Worked example

p q ¬ p ¬ q p ∨ q (¬ p) ∧ (¬ q) (p ∨ q) ∧ [(¬ p) ∧ (¬ q)]

T T F F T F F

T F F T T F F

F T T T T F F

F F T T F T F

2 Copy and complete the truth table below for the three statements p, q and r.

Logical contradiction and tautologyLogical contradictionA contradiction or contradictory proposition is never true. For example, let p bethe proposition that Rome is in Italy.

p: Rome is in Italy.

Therefore ¬ p, the negation of p, is the proposition: Rome is not in Italy.

If we write p ∧ ¬ p we are saying Rome is in Italy and Rome is not in Italy. Thiscannot be true at the same time. This is an example of a logical contradiction.

A truth table is shown below for the above statement.

Both entries in the final column are F. In other words a logical contradiction mustbe false.

Show that the compound proposition below is a contradiction.

(p ∨ q) ∧ [(¬ p) ∧ (¬ q)]

Construct a truth table:

Because the entries in the last column are all false, the statement is a logicalcontradiction.


p q r ¬ p p ∨ q ¬ p ∨ r (p ∨ q) ∧ (¬ p ∨ r)

T T T F T T T

T T F

T

T

F

F

F

F

p ¬ p p ∧ ¬ p

T F F

F T F


Truth tables 83

TautologyThe manager of the band Muse said to me recently: ‘If Muse’s album “Resistance” isa success, they will be a bigger band than U2.’ He paused ‘Or they will not’.

This is an example of a tautology: ‘either it does or it doesn’t’. It is always true.

A compound proposition is a tautology if it always true regardless of the truthvalues of its variables.

Consider the proposition: All students study maths or all students do not studymaths. This is a tautology, as can be shown in a truth table by considering theresult of p ∨ ¬ p.

Since the entries in the final column p ∨ ¬ p are all true, this is a tautology.

Show that (p ∨ q) ∨ [(¬ p) ∧ (¬ q)] is a tautology by copying and completing thetruth table below.

As the entries in the final column (p ∨ q) ∨ [(¬ p) ∧ (¬ q)] are all true, thestatement is a tautology.

� Exercise 3.6.21 Describe each of the following as a tautology, a contradiction or neither. Use a

truth table if necessary.a p ∧ ¬ qb q ∧ ¬ qc p ∨ ¬ qd q ∨ ¬ qe [p ∨ (¬ q)] ∧ [q ∨ (¬ q)]

2 By drawing a truth table in each case, decide whether each of the followingpropositions is a tautology, contradiction or neither.

a ¬ p ∧ ¬ qb ¬ (¬ p) ∨ pc q ∧ ¬ rd (p ∧ q) ∧ re (p ∧ q) ∨ r

p ¬ p p ∨ ¬ p

T F T

F T T

Worked example

p q ¬ p ¬ q p ∨ q (¬ p) ∧ (¬ q) (p ∨ q) ∨ [(¬ p) ∧ (¬ q)]

T T F F T F T

T F F T T F T

F T T F T F T

F F T T F T T



3.7 Implication; converse; inverse;contrapositive and logical equivalenceImplication‘If ’ is a word introducing a conditional clause. Later in your life someone might say to you, ‘If you get a degree, then I will buy youa car’.

Let us look at this in a truth table.

p: You get a degree.q: I will buy you a car.

The first row is simple:

You get a degree, I buy you a car, and therefore I have kept my promise.

The second row too is straightforward:

You get a degree, I don’t buy you a car, and therefore I have broken my promise.

The last two rows seem more complicated, but think of them like this. If you do notget a degree, then I have kept my side of the bargain whether I buy you a car or not.

Therefore, the only way that this type of statement is false is if a ‘promise’ isbroken.

Logically p ⇒ q is true if:

p is falseor q is trueor p is false and q is true

Similarly p ⇒ q is only false if p is true and q is false.

In the following statements, assume that the first phrase is p and the second phrase q.

Determine whether the statement p ⇒ q is logically true or false.

1 ‘If 5 × 4 = 20, then the Earth moves round the Sun.’As both p and q are true, then p ⇒ q is true, i.e. the statement p ⇒ q is logicallycorrect.

2 ‘If the Sun goes round the Earth, then I am an alien.’Since p is false, then p ⇒ q is true whether I am an alien or not. Therefore thestatement is logically true.

This means that witty replies like:

‘If I could run faster, I could be a professional footballer’‘Yes and if you had wheels you’d be a professional skater’ are logically true, sincethe premise p, ‘if you had wheels’, is false and therefore what follows is irrelevant.

p q p ⇒ q

T T T

T F F

F T T

F F T

Worked examples


Implication; converse; inverse; contrapositive and logical equivalence 85

� Exercise 3.7.11 In the following statements, assume that the first phrase is p and the second

phrase is q. Determine whether the statement p ⇒ q is logically true or false.a If 2 + 2 = 5 then 2 + 3 = 5.b If the moon is round, then the Earth is flat.c If the Earth is flat, then the moon is flat.d If the Earth is round, then the moon is round.e If the Earth is round, then I am the man on the moon.

2 Descartes’ phrase ‘Cogito, ergo sum’ translates as ‘I think, therefore I am’.a Rewrite the sentence using one or more of the following: ‘if’, ‘whenever’ ,

‘it follows that’, ‘it is necessary’ , ‘unless’ , ‘only’.b Copy and complete the following sentence: ‘Cogito ergo sum’ only breaks

down logically if Descartes thinks, but . . .’

Logical equivalenceThere are many different ways that we can form compound statements from thepropositions p and q using connectives. Some of the different compoundpropositions have the same truth values. These propositions are said to beequivalent. The symbol for equivalence is ⇔.

Two propositions are logically equivalent when they have identical truth values.

Use a truth table to show that ¬ (p ∨ q) and ¬ p ∨ ¬ q are logically equivalent.

Since the truth values for ¬ (p ∧ q) and ¬ p ∨ ¬ q (columns 4 and 7) are identical,the two statements are logically equivalent.

ConverseThe statement ‘All squares are rectangles’ can be rewritten using the word ‘if’ as:

‘If an object is a square, then it is a rectangle’. p ⇒ q. (true in this case)

The converse is:

q ⇒ p. ‘If an object is a rectangle, then it is a square.’ (false in this case)

InverseThe inverse of the statement ‘If an object is a square, then it is a rectangle’ (p ⇒ q) is:

¬p ⇒ ¬ q. ‘If an object is not a square, then it is not a rectangle.’ (false in this case)

p q p ∧ q ¬ (p ∧ q) ¬ p ¬ q ¬ p ∨ ¬ q

T T T F F F F

T F F T F T T

F T F T T F T

F F F T T T T

Worked example


Implication Contrapositivep q ¬p ¬q p ⇒ q ¬q ⇒ ¬p

T T F F T T

T F F T F F

F T T F T T

F F T T T T

ContrapositiveThe contrapositive of the statement ‘If an object is a square, then it is a rectangle’(p ⇒ q) is:

¬ q ⇒ ¬ p. ‘If an object is not a rectangle, then it is not a square.’ (true in this case)

Note: A statement is logically equivalent to its contrapositive. A statement is not logically equivalent to its converse or inverse.The converse of a statement is logically equivalent to the inverse.

So if a statement is true, then its contrapositive is also true.If a statement is false, then its contrapositive is also false.

And if the converse of a statement is true, then the inverse is also true.If the converse of a statement is false, then the inverse is also false.

To summarize: given a conditional statement: p ⇒ qthe converse is: q ⇒ pthe inverse is: ¬ p ⇒ ¬ qthe contrapositive is: ¬ q ⇒ ¬ p

Statement: ‘All even numbers are divisible by 2.’a Rewrite the statement as a conditional statement. b State the converse, inverse and contrapositive of the conditional

statement. State whether each new statement is true or false.

a Conditional: ‘If a number is even, then it is divisible by 2.’ (true)b Converse: ‘If a number is divisible by 2, then it is an even number.’ (true)

Inverse: ‘If a number is not even, then it is not divisible by 2.’ (true)Contrapositive: ‘If a number is not divisible by 2, then it is not an evennumber. (true)

Note: The contrapositive both switches the order and negates. It combines theconverse and the inverse.

On a truth table it can be shown that a conditional statement and itscontrapositive are logically equivalent

Note: If we have a tautology, we must have logical equivalence. For example,‘If you cannot find the keys you have lost, then you are looking in the wrong place.’Obviously if you are looking in the right place then you can find your keys. (So thecontrapositive is equivalent to the proposition.)


Worked example


� Exercise 3.7.21 Write each of the following as a conditional statement and then write its

converse, inverse or contrapositive, as indicated in brackets.

Example Being interested in the Romans means that you will enjoy Italy.(converse)

Solution: Conditional statement. If you are interested in the Romans, then youwill enjoy Italy.

Converse.If you enjoy Italy, then you are interested in the Romans.

a You do not have your mobile phone, so you cannot send a text. (inverse)b A small car will go a long way on 20 euros worth of petrol. (contrapositive)c Speaking in French means that you will enjoy France more. (converse)d When it rains I do not play tennis. (inverse) e We stop playing golf when there is a threat of lightning. (inverse)f The tennis serve is easy if you practise it. (contrapositive)g A six-sided polygon is a hexagon. (contrapositive)h You are less than 160 cm tall, so you are smaller than me. (inverse)i The bus was full, so I was late. (contrapositive)j The road was greasy, so the car skidded. (converse)

2 Rewrite these statements using the conditional ‘if’. Then state the converse,inverse and contrapositive. State whether each new statement is true or false.

a Any odd number is a prime number.b A polygon with six sides is called an octagon.c An acute-angled triangle has three acute angles.d Similar triangles are congruent.e Congruent triangles are similar.f A cuboid has six faces.g A solid with eight faces is a regular octahedron.h All prime numbers are even numbers.

3.8 Probability Although Newton and Galileo had some thoughts about chance, it is accepted thatthe study of what we now call probability began when Blaise Pascal (1623−1662)and Pierre de Fermat (of Fermat’s last theorem fame) corresponded about problemsconnected with games of chance. Later Christiaan Huygens wrote the first book onthe subject, The Value of all Chances in Games of Fortune, in 1657. This included achapter entitled ‘Gambler’s Ruin’.

In 1821 Carl Friedrich Gauss (1777−1855), one of the greatest mathematicianswho ever lived, worked on the ‘Normal distribution’, a very important contributionto the study of probability.

Probability is the study of chance, or the likelihood of an event happening. Inthis section we will be looking at theoretical probability. But, because probability isbased on chance, what theory predicts does not necessarily happen in practice.

Probability 87

Pierre de Fermat


Probability of an eventA favourable outcome refers to the event in question actually happening. The totalnumber of possible outcomes refers to all the different types of outcome one can getin a particular situation. In general:

Probability of an event =

This can also be written as: P(A) = ,

where P(A) is the probability of event A, n(A) is the number of ways event A canoccur and n(U) is the total number of equally likely outcomes.

Therefore

if the probability = 0, it implies the event is impossibleif the probability = 1, it implies the event is certain to happen

An ordinary, fair dice is rolled. a Calculate the probability of getting a 6.b Calculate the probability of not getting a 6.

a Number of favourable outcomes = 1 (i.e. getting a 6)

Total number of possible outcomes = 6 (i.e. getting a 1, 2, 3, 4, 5 or 6)

Probability of getting a 6, P(6) =

b Number of favourable outcomes = 5 (i.e. getting a 1, 2, 3, 4, 5)Total number of possible outcomes = 6 (i.e. getting a 1, 2, 3, 4, 5 or 6)

Probability of not getting a six, P(6′) =

From this it can be seen that the probability of not getting a 6 is equal to 1minus the probability of getting a 6, i.e. P(6) = 1 − P(6′).

These are known as complementary events.

In general, for an event A, P(A) = 1 − P(A′).

� Exercise 3.8.11 Calculate the theoretical probability, when rolling an ordinary, fair dice, of

getting each of the following.a a score of 1 b a score of 5c an odd number d a score less than 6e a score of 7 f a score less than 7

2 a Calculate the probability of:i) being born on a Wednesdayii) not being born on a Wednesday.

b Explain the result of adding the answers to a i) and ii) together.

56

16

n(A)n(U)

number of favourable outcomestotal number of equally likely outcomes

Worked example



Probability 89

3 250 tickets are sold for a raffle. What is the probability of winning if you buy:a 1 ticket b 5 ticketsc 250 tickets d 0 tickets?

4 In a class there are 25 girls and 15 boys. The teacher takes in all of their booksin a random order. Calculate the probability that the teacher will:

a mark a book belonging to a girl firstb mark a book belonging to a boy first.

5 Tiles, each lettered with one different letter of the alphabet, are put into a bag.If one tile is drawn out at random, calculate the probability that it is:

a an A or P b a vowelc a consonant d an X, Y or Z.e a letter in your first name.

6 A boy was late for school 5 times in the previous 30 school days. If tomorrow isa school day, calculate the probability that he will arrive late.

7 3 red, 10 white, 5 blue and 2 green counters are put into a bag. a If one is picked at random, calculate the probability that it is:

i) a green counterii) a blue counter.

b If the first counter taken out is green and it is not put back into the bag,calculate the probability that the second counter picked is:i) a green counterii) a red counter.

8 A roulette wheel has the numbers 0 to 36 equally spaced around its edge. Assuming that it is unbiased, calculate the probability on spinning it of getting:

a the number 5 b an even numberc an odd number d zeroe a number greater than 15 f a multiple of 3g a multiple of 3 or 5 h a prime number.

9 The letters R, C and A can be combined in several different ways.a Write the letters in as many different combinations as possible.b If a computer writes these three letters at random, calculate the

probability that:i) the letters will be written in alphabetical orderii) the letter R is written before both the letters A and Ciii) the letter C is written after the letter Aiv) the computer will spell the word CART if the letter T is added.

10 A normal pack of playing cards contains 52 cards. These are made up of foursuits (hearts, diamonds, clubs and spades). Each suit consists of 13 cards. Theseare labelled ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen and King. The hearts anddiamonds are red; the clubs and spades are black.

If a card is picked at random from a normal pack of cards calculate theprobability of picking:

a a heart b a black cardc a four d a red Kinge a Jack, Queen or King f the ace of spadesg an even numbered card h a seven or a club.


3.9 Combined eventsIn this section we look at the probability of two or more events happening:combined events. If only two events are involved, then two-way tables can be used to show the outcomes.

Two-way tables of outcomes

a Two coins are tossed. Show all the possible outcomes in a two-way table.b Calculate the probability of getting two heads.c Calculate the probability of getting a head and a tail in any order.

a

b All four outcomes are equally likely, therefore the probability of getting HH is .

c The probability of getting a head and a tail in any order, i.e. HT or TH, is = .

� Exercise 3.9.11 a Two fair tetrahedral dice are rolled. If each is numbered 1−4, draw a two-way

table to show all the possible outcomes.b What is the probability that both dice show the same number?c What is the probability that the number on one dice is double the number on

the other?d What is the probability that the sum of both numbers is prime?

2 Two fair dice are rolled. Copy and complete the diagram to show all the possiblecombinations.

What is the probability of getting:a a double 3b any doublec a total score of 11d a total score of 7e an even number on both dicef an even number on at least one diceg a 6 or a doubleh scores which differ by 3i a total which is either a multiple of 2 or 5?

12

24

14

Worked example

Head Tail

Hea

dTa

il

Coin 1

HH TH

HT TT

Co

in 2

6

5

4

3

2

1

3,6

3,5

3,4

3,3

3,2

Dic

e 2

Dice 1

1,1 2,1 3,1 4,1

6,25,2

1 2 3 4 5 6



Combined events 91

Tree diagramsWhen more than two combined events are being considered, two-way tables cannotbe used and therefore another method of representing information diagrammaticallyis needed. Tree diagrams are a good way of doing this.

a If a coin is tossed three times, show all the possible outcomes on a tree diagram,writing each of the probabilities at the side of the branches.

b What is the probability of getting three heads? c What is the probability of getting two heads and one tail in any order?d What is the probability of getting at least one head?e What is the probability of getting no heads?

a

b To calculate the probability of getting three heads, multiply along the branches:

P(HHH) = × × =

c The successful outcomes are HHT, HTH, THH.P(two heads, one tail any order)

= P(HHT) + P(HTH) + P(THH)

= ( × × ) + ( × × ) + ( × × ) = + + =

Therefore the probability is .

d This refers to any outcome with either one, two or three heads, i.e. all of themexcept TTT.

P(TTT) = × × =

P(at least one head) = 1 − P(TTT) = 1 − =

Therefore the probability is .

e The only successful outcome for this event is TTT.

Therefore the probability is , as shown in part d.18

78

18

78

12

12

12

18

38

18

18

18

38

12

12

12

12

12

12

12

12

12

12

12

12

18

Worked example

H

HH

T

T

H

T

T

HH

T

T

H

T

�� HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

OutcomesToss 1 Toss 2 Toss 3

��

��

��

��

��

��

��

��

��

��

��

��

��


� Exercise 3.9.21 a A computer uses the numbers 1, 2 and 3 at random to make three-digit

numbers. Assuming that a number can be repeated, show on a tree diagram allthe possible combinations that the computer can print.

b Calculate the probability of getting:i) the number 131 ii) an even numberiii) a multiple of 11 iv) a multiple of 3v) a multiple of 2 or 3 vi) a palindromic number.

2 a A family has four children. Draw a tree diagram to show all the possiblecombinations of boys and girls. [Assume P(girl) = P(boy).]

b Calculate the probability of getting:i) all girls ii) two girls and two boysiii) at least one girl iv) more girls than boys.

3 a A netball team plays three matches. In each match the team is equally likelyto win, lose or draw. Draw a tree diagram to show all the possible outcomesover the three matches.

b Calculate the probability that the team:i) wins all three matchesii) wins more times than it losesiii) loses at least one matchiv) doesn’t win any of the three matches.

c Explain why it is not very realistic to assume that the outcomes are equallylikely in this case.

4 A spinner is split into quarters.

a If it is spun twice, draw a probability tree showing all the possible outcomes.b Calculate the probability of getting:

i) two greensii) a green and a blue in any orderiii) a blue and a white in any order.

Tree diagrams for unequal probabilitiesIn each of the cases considered so far, all of the outcomes have been assumed to beequally likely. However, this need not be the case.



Combined events 93

In winter, the probability that it rains on any one day is .

a Using a tree diagram, show all the possible combinations for two consecutivedays. Write each of the probabilities by the sides of the branches.

b Calculate the probability that it will rain on both days.c Calculate the probability that it will rain on the first day but not the second day.d Calculate the probability that it will rain on at least one day.

a

Note how the probability of each outcome is found by multiplying theprobabilities for each of the branches.

b P(R, R) = × =

c P(R, NR) = × =

d The outcomes which satisfy this event are (R, R), (R, NR) and (NR, R).

Therefore the probability is + + =

� Exercise 3.9.31 A particular board game involves players rolling a dice. However, before a player

can start, he or she needs to roll a 6.a Copy and complete the tree diagram below showing all the possible

combinations for the first two rolls of the dice.

57

2549

1049

1049

4549

57

27

1049

57

57

2549

Worked example

��

Rain

Rain

No rain

No rain

Rain

No rain

Day 1 Day 2 Outcomes

Rain, Rain

Rain, No rain

No rain, Rain

No rain, No rain

��

� �

��

� �

Probability

��

��

��

��

��

��

��

��

Roll 1 Roll 2 Outcomes Probability

��

��

Six, SixSix

Not six

Six

Not six

Six

Not six

536–


b Calculate the probability of each of the following.i) Getting a six on the first throwii) Starting within the first two throwsiii) Starting on the second throwiv) Not starting within the first three throwsv) Starting within the first three throws

c If you add the answers to b iv) and v) what do you notice? Explain.

2 In Italy of the cars are foreign made. By drawing a tree diagram and writing the

probabilities next to each of the branches, calculate each of these probabilities.a The next two cars to pass a particular spot are both Italian.b Two of the next three cars are foreign.c At least one of the next three cars is Italian.

3 The probability that a morning bus arrives on time is 65%.a Draw a tree diagram showing all the possible outcomes for three

consecutive mornings.b Label your tree diagram and use it to calculate the probability of each of

the following.i) The bus is on time on all three mornings.ii) The bus is late the first two mornings.iii) The bus is on time two out of the three mornings.iv) The bus is on time at least twice.

4 Light bulbs are packaged in cartons of three; 10% of the bulbs are found to befaulty. Calculate the probability of finding two faulty bulbs in a single carton.

5 A volleyball team has a 0.25 chance of losing a game. Calculate the probabilityof the team achieving:

a two consecutive wins b three consecutive winsc 10 consecutive wins.

Tree diagrams for probability problems with and without ‘replacement’In the examples considered so far, the probability for each outcome remained thesame throughout the problem. However, this need not always be the case.

1 A bag contains three red balls and seven black balls. If the balls are put backafter being picked, what is the probability of picking:

a two red ballsb a red ball and a black ball in any order.

This is selection with replacement. Draw atree diagram to help visualise the problem.

35

redred

black

black

red

black

310

710

710

310

710

310

Worked examples



a The probability of a red followed by a red, P(RR) = × = .

b The probability of a red followed by a black or a black followed by a red is

P(RB) + P(BR) = ( × ) + ( × ) = + = .

2 Repeat question 1, but this time each ball that is picked is not put back in thebag.

This is selection without replacement. The tree diagram is now as shown.

a P(RR) = × = .

b P(RB) + P(BR) = ( × ) + ( × ) = + = .

� Exercise 3.9.41 A bag contains five red balls and four black balls. If a ball is picked out at

random, its colour is recorded and it is then put back in the bag, what is theprobability of choosing:

a two red ballsb two black ballsc a red ball and a black ball in this orderd a red ball and a black ball in any order?

2 Repeat question 1 but, in this case, after a ball is picked at random, it is not putback in the bag.

3 A bag contains two black, three white and five red balls. If a ball is picked, its colour recorded and then put back in the bag, what is the probability of picking:

a two black ballsb a red and a white ball in any order?

4 Repeat question 3 but, in this case, after a ball is picked at random, it is not putback in the bag.

5 You buy five tickets for a raffle. 100 tickets are sold altogether. Tickets are pickedat random. You have not won a prize after the first three tickets have been drawn.

a What is the probability that you win a prize with either of the next twodraws?

b What is the probability that you do not win a prize with either of the nexttwo draws?

310

310

9100

310

79

710

39

2190

2190

4290

310

29

690

310

710

710

310

21100

21100

42100

Combined events 95

redred

black

black

red

black

310

710

69

39

79

29


6 A bowl of fruit contains one apple, one banana, two oranges and two pears. Twopieces of fruit are chosen at random and eaten.

a Draw a probability tree showing all the possible combinations of the twopieces of fruit.

b Use your tree diagram to calculate the probability that:i) both the pieces of fruit eaten are orangesii) an apple and a banana are eateniii) at least one pear is eaten.

Use of Venn diagrams in probabilityYou have seen earlier in this unit how Venn diagrams can be used to represent sets.They can also be used to solve problems involving probability.

Probability of event, A, P(A) =

1 In a survey carried out in a college, students were asked for their favouritesubject.

15 chose English8 chose Science12 chose Mathematics5 chose Art

If a student is chosen at random, what is the probability that he or she likesScience best?

This can be represented on a Venn diagram as:

There are 40 students, so the probability is = .

2 A group of 21 friends decide to go out for the day to the local town; 9 of themdecide to see a film at the cinema, 15 of them get together for lunch.

a Draw a Venn diagram to show this information if set A represents thosewho see a film and set B those who have lunch.

b Determine the probability that a person picked at random only went to thecinema.

840

15

n(A)n(U)

Worked examples

English15 Maths

12Art5

Science8



Laws of probability 97

a 9 + 15 = 24; as there are only 21 people,this implies that 3 people see the film andhave lunch. This means that 9 − 3 = 6 only went to see a film and 15 − 3 = 12 only had lunch.

b The number who only went to the cinema is 6, as the other 3 who saw a film also went out for lunch. Therefore the probability is = .

� Exercise 3.9.51 In a class of 30 students, 20 study French, 18 study Spanish and 5 study neither.

a Draw a Venn diagram to show this information.b What is the probability that a student chosen at random studies both

French and Spanish?

2 In a group of 35 students, 19 take Physics, 18 take Chemistry and 3 take neither.What is the probability that a student chosen at random takes:

a both Physics and Chemistryb Physics onlyc Chemistry only?

3 108 people visited an art gallery; 60 liked the pictures, 53 liked the sculpture, 10liked neither.

What is the probability that a person chosen at random liked the pictures butnot the sculpture?

4 In a series of examinations in a school:37 students took English48 students took French45 students took Spanish15 students took English and French13 students took French and Spanish7 students took English and Spanish5 students took all three.a Draw a Venn diagram to represent this information.b What is the probability that a student picked at random took:

i) all threeii) English onlyiii) French only?

3.10 Laws of probabilityMutually exclusive eventsEvents that cannot happen at the same time are known as mutually exclusiveevents. For example, if a sweet bag contains 12 red sweets and 8 yellow sweets, letpicking a red sweet be event A, and picking a yellow sweet be event B. If one sweetis picked, it is not possible to pick a sweet which is both red and yellow. Thereforethese events are mutually exclusive.

621

27

6 3 12

A BU



This can be shown in a Venn diagram:

P(A) = whilst P(B) = .

As there is no overlap, P(A � B) = P(A) + P(B) = + = = 1.

i.e. the probability of mutually exclusive event A or event B happening is equal tothe sum of the probabilities of event A and event B and the sum of the probabilitiesof all possible mutually exclusive events is 1.

In a 50 m swim, the world record holder has a probability of 0.72 of winning. Theprobability of her finishing second is 0.25.

What is the probability that she either wins or comes second?

Since she cannot finish both first and second, the events are mutually exclusive.Therefore P(1st � 2nd) = 0.72 + 0.25 = 0.97.

Combined eventsIf events are not mutually exclusive then they may occur at the same time.

These are known as combined events.

For example, a pack of 52 cards contains four suits: clubs (♣), spades (♠), hearts(♥) and diamonds (♦). Clubs and spades are black; hearts and diamonds are red.Each suit contains 13 cards. These are ace, 2, 3, 4, 5, 6, 7, 8, 9,10, Jack, Queen andKing.

A card is picked at random. Event A represents picking a black card; event Brepresents picking a King.

In a Venn diagram this can be shown as:

P(A) = = and P(B) = =452

113

2652

14

1220

820

2020

1220

820

A BU

Worked example

UA♥

A BA♠ 2♠ 3♠4♠ 5♠ 6♠7♠ 8♠ 9♠

10♠ J♠ Q♠

A♣ 2♣ 3♣4♣ 5♣ 6♣7♣ 8♣ 9♣

10♣ J♣ Q♣

K♠

K♣

K♥

K♦

2♥ 3♥ 4♥ 5♥ 6♥ 7♥ 8♥ 9♥ 10♥ J♥ Q♥

A♦ 2♦ 3♦ 4♦ 5♦ 6♦ 7♦ 8♦ 9♦ 10♦ J♦ Q♦


However P(A � B) ≠ + because K♠ and K♣ belong to both events A and Band have therefore been counted twice. This is shown in the overlap of the Venndiagram.

Therefore, for combined events, P(A � B) = P(A) + P(B) − P(A � B)

i.e. the probability of event A or B is equal to the sum of the probabilities of A andB minus the probability of A and B.

In a holiday survey of 100 people:

72 people have had a beach holiday 16 have had a skiing holiday12 have had both.

What is the probability that one person chosen at random from the survey has hadeither a beach holiday (B) or a ski holiday (S)?

P(B) = P(S) = P(B � S) =

Therefore P(B � S) = + − =

Independent eventsA student may be born on 1 June, another student in his class may also be born on1 June. These events are independent of each other (assuming they are not twins).

If a dice is rolled and a coin spun, the outcomes of each are also independent,i.e. the outcome of one does not affect the outcome of another.

For independent events, the probability of both events occurring is the productof each occurring separately, i.e.

P(A � B) = P(A) × P(B)

1 I spin a coin and roll a dice.a What is the probability of getting a head on the coin and a five on the dice?b What is the probability of getting either a head on the coin or a five on

the dice, but not both?

a P(H) = P(5) =

Both events are independent therefore P(H � 5) = P(H) × P(5)

= ×

=

b P(H � 5) is the probability of getting a head, a five or both.Therefore P(H � 5) − P(H � 5) removes the probability of both eventsoccurring.The solution is P(H � 5) − P(H � 5) = P(H) + P(5) − P(H � 5)

= + −

= 712

12

16

112

112

12

16

12

16

72100

16100

12100

76100

72100

16100

12100

2652

452


Worked example

Worked examples


2 The probabilities of two events X and Y are given by:

P(X) = 0.5, P(Y) = 0.4, and P(X � Y) = 0.2.

a Are events X and Y mutually exclusive?b Calculate P(X � Y).c What kind of events are X and Y?

a No: if the events were mutually exclusive, then P(X � Y) would be 0 asthe events could not occur at the same time.

b P(X � Y) = P(X) + P(Y) − P(X � Y)= 0.5 + 0.4 − 0.2 = 0.7

c Since P(X � Y)= P(X) × P(Y), i.e.0.2 = 0.5 × 0.4, events X and Y must be independent.

Conditional probabilityConditional probability refers to the probability of an event (A) occurring, which isin turn dependent on another event (B).

For example, a group of ten children play two tennis matches each. The tablebelow shows which matches the children won and lost.

Child First match Second match

1 Won Won

2 Lost Won

3 Lost Won

4 Won Lost

5 Lost Lost

6 Won Lost

7 Won Won

8 Won Won

9 Lost Won

10 Lost Won

Let winning the first match be event A and winning the second match be event B.An example of conditional probability would be as follows: calculate the probabilitythat a boy picked at random won his first match, if it is known that he won hissecond match.

Because we are told that the boy won his second match, this will affect the finalprobability.

This is written as P(A⎥ B), i.e. the probability of event A given that event B hashappened.

P(A⎥ B) =

or P(A⎥ B) = where n is the number of times that event happens.n(A � B)n(B)

P(A � B)P(B)




Using the table on the previous page, for a child picked at random:

a Calculate the probability that the child lost both matches.b Calculate the probability that the child won his first match.c Calculate the probability that a child won his first match, if it is known that he

won his second match.

a P(A′ � B′) = =

b P(A) = = =

c P(A⎥ B) = =

Note: The answers to b and c are different although both relate to the probabilityof a child winning his first match.

Venn diagrams are very useful for conditional probability as they show n(A � B)clearly.

In a class of 25 students, 18 play football, 8 play tennis and 6 play neither sport.

a Show this information on a Venn diagram.b What is the probability that a student chosen at random plays both sports?c What is the probability that a student chosen at random plays football, given

that he also plays tennis?

a 18 + 8 + 6 = 32. As there are only 25 students, 7 must play both sports.

b P(F � T) = =

c P(F ⎥ T) = =

� Exercise 3.10.11 The Jamaican 100 m women’s relay team has a 0.5 chance of coming first in the

final, 0.25 chance of coming second and 0.05 chance of coming third. a Are the events independent?b What is the team’s chance of a medal?

78

n(F � T)n(T)

n(F � T)n(U)

725

n(A � B)n(B)

37

n(A)n(U)

510

12

n(A′ � B′)n(U)

110

Worked example

Worked example

11

F T

7 1

6

U


2 I spin a coin and throw a dice. a Are the events independent?b What is the probability of getting:

i) a head and a factor of 3ii) a head or a factor of 3iii) a head or a factor of 3, but not both?

3 What is the probability that two people picked at random both have a birthdayin June?

4 Amelia takes two buses to work. On a particular day, the probability of hercatching the first bus is 0.7 and the probability of her catching the second bus is0.5. The probability of her catching neither is 0.1.

a Are the events independent?b If A represents catching the first bus and B the second:

i) State P(A � B)′.ii) Find P(A � B).iii) Given that P(A � B) = P(A) + P(B) − P(A � B), calculate

P(A � B).iv) Calculate the probability P(A⎥ B), i.e. the probability of Amelia

having caught the first bus, given that she caught the second bus.

5 The probability of Marco having breakfast is 0.75. The probability that he gets alift to work is 0.9 if he has had breakfast and 0.8 if he has not.

a What is the probability of Marco having breakfast then getting a lift?b What is the probability of Marco not having breakfast then getting a lift?c What is the probability that Marco gets a lift?d If Marco gets a lift, what is the probability that he had breakfast?

6 Inês has a driving test on Monday and a Drama exam the next day. Theprobability of her passing the driving test is 0.73. The probability of her passingthe Drama exam is 0.9. The probability of failing both is 0.05.

Given that she has passed the driving test, what is the probability that she alsopassed her Drama exam?

7 An Olympic swimmer has a 0.6 chance of a gold medal in the 100 m freestyle, a0.7 chance of a gold medal in the 200 m freestyle and a 0.1 chance of no goldmedals. Given that she wins the 100 m race, what is the probability of herwinning the 200 m race?

8 a How many pupils are in your class?b How likely do you think it is that two people in your class will share the same

birthday? Very likely? Likely? Approx 50−50? Unlikely? Very unlikely? c Write down everybody’s birthday. Did two people have the same birthday?

Below is a way of calculating the probability that two people have the samebirthday depending on how many people there are. To study this it is easiestto look at the probability of birthdays being different. When this probability isless than 50%, then the probability that two people will have the samebirthday is greater than 50%.

When the first person asks the second person, the probability of them not having the same birthday is (i.e. it is that they have the same birthday).

364365

1365



When the next person is asked, as the events are independent, the probabilityof all three having different birthdays is:

( ) × ( ) = 99.2%

When the next person is asked, the probability of all four having differentbirthdays is:

( ) × ( ) × ( ) = 98.4%

and so on….

d Copy and complete the table below until the probability is 50%.

Number of people Probability of them not having the same birthday

3642 ––––– = 99.7%365

364 3633 (–––––) × (–––––) = 99.2%365 365

364 363 3624 (–––––) × (–––––) × (–––––) = 98.4%365 365 365

5

10

15

20

etc.

e Explain in words what your solution to part d means.

364365

363365

362365

364365

363365

Student assessments 103

� Student assessment 11 Describe the following sets in words.

a {1, 3, 5, 7}b {1, 3, 5, 7, ...}c {1, 4, 9, 16, 25, ...}d {Arctic, Atlantic, Indian, Pacific}

2 Calculate the value of n(A) for each of the setsshown below.a A is the set of days of the weekb A is the set of prime numbers between 50

and 60c A = {x⎥ x is an integer and −9 ≤ x ≤ −3}d A is the set of students in your class

3 Copy this Venn diagram three times.

a On one copy shade and label the regionwhich represents A � B.

b On another copy shade and label the regionwhich represents A � B.

c On the third copy shade and label the regionwhich represents (A � B)′.

4 If A = {w, o, r, k}, list all the subsets of A withat least three elements.

AU

B


5 If U = {1, 2, 3, 4, 5, 6, 7, 8} and P = {2, 4, 6, 8},what set is represented by P′?

6 A hexagonal spinner is divided into equilateraltriangles painted alternately red and black.What is the sample space when the spinner isspun three times?

7 If p is the proposition ‘The Amazon river is in Africa’, write the proposition ¬ p in words.

8 What is meant by p ∨ q?

9 Calculate the theoretical probability of:a being born on a Saturdayb being born on the 5th of a month in a non-

leap yearc being born on 20 June in a non-leap yeard being born on 29 February.

10 A coin is tossed and an ordinary, fair dice isrolled.a Draw a two-way table showing all the

possible combinations.b Calculate the probability of getting:

i) a head and a 6ii) a tail and an odd numberiii) a head and a prime number.

� Student assessment 21 If A = {2, 4, 6, 8}, write all the proper subsets of

A with two or more elements.

2 X = {lion, tiger, cheetah, leopard, puma, jaguar,cat}Y = {elephant, lion, zebra, cheetah, gazelle}Z = {anaconda, jaguar, tarantula, mosquito}a Draw a Venn diagram to represent the above

information.b Copy and complete the statement

X � Y = {...}.c Copy and complete the statement

Y � Z = {...}.d Copy and complete the statement

X � Y � Z = {...}.

3 U is the set of natural numbers, M is the set ofeven numbers and N is the set of multiples of 5.a Draw a Venn diagram and place the numbers

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 in the appropriateplaces in it.

b If X = M � N, describe set X in words.

4 A group of 40 people were asked whether theylike tennis (T) and football (F). The numberliking both tennis and football was three timesthe number liking only tennis. Adding 3 to thenumber liking only tennis and doubling theanswer equals the number of people liking onlyfootball. Four said they did not like sport at all.a Draw a Venn diagram to represent this

information.b Calculate n(T � F).c Calculate n(T � F′).d Calculate n(T′ � F).

5 The Venn diagram below shows the number ofelements in three sets P, Q and R.

If n(P � Q � R) = 93 calculate:a xb n(P)c n(Q)d n(R)e n(P � Q)f n(Q � R)g n(P � R)h n(R � Q)i n(P � Q)′.

6 What is meant by p ∧ q?

7 Copy and complete the truth table below.

P Q

R

28 �x

12 �x10 �x

x

10 �x 15 �x 13 �x

p q p ∧ q p ∨ q

T

T

F

F



Student assessments 105

8 What is a tautology? Give an example.

9 A goalkeeper expects to save one penalty out ofevery three. Calculate the probability that he:a saves one penalty out of the next threeb fails to save any of the next three penaltiesc saves two out of the next three penalties.

� Student assessment 31 The probability that a student takes English is

0.8. The probability that a student takes Englishand Spanish is 0.25.

What is the probability that a student takesSpanish, given that he takes English?

2 A card is drawn from a standard pack of cards. a Draw a Venn diagram to show the following:

A is the set of acesB is the set of picture cardsC is the set of clubs

b From your Venn diagram find the followingprobabilities.i) P(ace or picture card)ii) P(not an ace or picture card)iii) P(club or ace)iv) P(club and ace)v) P(ace and picture card)

3 Students in a school can choose to study one ormore science subjects from physics, chemistryand biology.

In a year group of 120 students, 60 tookphysics, 60 took biology and 72 took chemistry;34 took physics and chemistry, 32 tookchemistry and biology and 24 took physics andbiology; 18 took all three. a Draw a Venn diagram to represent this

information.b If a student is chosen at random, what is the

probability that: i) the student chose to study only one

subjectii) the student chose physics or chemistry,

and did not choose biology?

4 A class took an English test and a maths test.40% passed both tests and 75% passed theEnglish test.

What percentage of those who passed theEnglish test also passed the maths test?

5 A jar contains blue and red counters. Twocounters are chosen without replacement. Theprobability of choosing a blue then a redcounter is 0.44. The probability of choosing ablue counter on the first draw is 0.5.

What is the probability of choosing a redcounter on the second draw if the first counterchosen was blue?

6 In a group of children, the probability that achild has black hair is 0.7. The probability thata child has brown eyes is 0.55. The probabilitythat a child has either black hair or brown eyesis 0.85.

What is the probability that a child chosen atrandom has both black hair and brown eyes?

7 A ball enters a chute at X.

a What are the probabilities of the ball goingdown each of the chutes labelled (i), (ii)and (iii)?

b Calculate the probability of the ball landingin:i) tray Aii) tray Ciii) tray B.

X

0.2

0.30.7

(i)

(ii)(iii)

A B C


Prevailing A head 106

3Discussion points, project ideas and theory of knowledge

Top

ic

4 Draw a Venn diagram torepresent Belief, Truth andKnowledge. Discuss thestatement ‘Knowledge isfound where belief andtruth intersect.’

3 Set theory is an area that

could be studied as a

project beyond the

Mathematical Studies

syllabus. Your teacher

may suggest some areas

of study.

5 The set of whole numbers

and the set of square

numbers have an infinite

number of elements. Does

this mean that there are

different values of infinity?

2 Research and discuss‘Bertrand’s Box Paradox’.This could be the startingpoint for a project onparadoxes.

6 What is the differencebetween zero and anempty set? Is a vacuum an empty set?

1 Research and discussRussell’s antinomy (not the element antimony).


Prevailing A head 107

10 I put out three cards face down, one

of which is an ace. I know the position of

the ace. You pick a card, but do not

turn it over. I then turn over a card

which is not the ace. You are then

offered the opportunity to change your

pick to the remaining card. Design a

probability experiment for many trials as

a class activity. Why does the probability

of you choosing the ace increase if you

make the change? How much does the

probability change by? This could be the

starting point for a project.

8 A teacher says that she

will give a surprise test on

one weekday of the following

week. Why can the test not

be a surprise if it is given on

the Friday? By extending that

reasoning, the test cannot be

given on Thursday either,

and so on. Discuss the fallacy

of this reasoning.

9 Is it possible to draw a Venndiagram on a two-dimensional space (a pieceof paper) to represent foursets which intersect at oneplace? (Think of U shapes).Is it possible to represent fivesets in a similar way? Thiscould make a possible

project.

11 What is a

paradox? ‘This

statement is a lie.’

Is that a paradox?

Discuss.

7 A pile of 100 000 grains of

sand is a heap. If sand is

removed one grain at a

time, at what point does

the pile cease to be a

heap?


Documents

Topic 3 Sets, logic and probability - kennso - home · PDF file · 2011-04-19Topic 3 Sets, logic and probability Syllabus content ... 2 A is the set of whole numbers between 50 and